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Let \(\mathfrak {F}\)  be a property of words. An \(\mathfrak {F}\) -factorisation of a word \(w\)  is a factorisation \(w = w_1\ldots w_n\)  such that each \(w_i\)  has the property \(\mathfrak {F}\) . Moreover \(\mathfrak {F}\)  is synchronising if every word has exactly one \(\mathfrak {F}\) -factorisation and additionally there exist \(l,r \in \mathbb {N}\)  such that for any words \(x,y\)  with \(\mathfrak {F}\) -factorisations \(x = x_1x_2\ldots x_s\)  and \(y = y_1 y_2 \ldots y_k\)  where \(k > l+r\)  and \(y\)  is a factor of \(x\) , the following hold:
